Uniform Asymptotic Expansion of the Voltage Potential in the Presence of Thin Inhomogeneities with Arbitrary Conductivity

Asymptotic expansions of the voltage potential in terms of the "radius" of a diametrically small(or several diametrically small) material inhomogeneity(ies) are by now quite well-known. Such asymptotic expansions for diametrically small inhomogeneities are uniform with respect to the conductivity of the inhomogeneities.In contrast, thin inhomogeneities, whose limit set is a smooth, codimension 1 manifold,σ, are examples of inhomogeneities for which the convergence to the background potential,or the standard expansion cannot be valid uniformly with respect to the conductivity, a, of the inhomogeneity. Indeed, by taking a close to 0 or to infinity, one obtains either a nearly homogeneous Neumann condition or nearly constant Dirichlet condition at the boundary of the inhomogeneity, and this difference in boundary condition is retained in the limit.The purpose of this paper is to find a "simple" replacement for the background potential, with the following properties:(1) This replacement may be(simply) calculated from the limiting domain Ω\σ, the boundary data on the boundary of Ω, and the right-hand side.(2) This replacement depends on the thickness of the inhomogeneity and the conductivity,a, through its boundary conditions on σ.(3) The difference between this replacement and the true voltage potential converges to 0 uniformly in a, as the inhomogeneity thickness tends to 0.     

混沌振荡系统的空时复杂度

基于相轨迹随时间的变化规律,提出了混沌振荡系统空时复杂度的概念,给出了空时复杂度的定义和计算方法.定义物理意义直观明确,与Lyapunov指数计算相比,方法计算量少,便于实际应用.以Duffing振子为例,通过数值仿真与实验,研究了混沌振荡系统的空时复杂度,实验结果表明空时复杂度可以很好地刻画Duffing振子丰富的动力学特性.     

Solitons in the Domain Structure of the Ferromagnet

By the method of dressing on a torus, we obtain and study solutions of the Landau–Lifshitz equation, which describe solitons in the stripe domain structure of the easy-axis ferromagnet. A specific feature of these solitons is that they are directly related to the domain structure: they induce translations and local oscillations of the domains. We find integrals of motion stabilizing the solitons on the background of the structure.     

Dynamics of Domain Walls in a Cylindrical Amorphous Ferromagnetic Microwire with Magnetic Inhomogeneities

Theoretical and Mathematical Physics - We study the dynamics of a domain wall (DW) in the magnetic core of thin amorphous glass-coated bistable microwires with a circular cross section containing...     

Internal Oscillations and Radiation Damping of Vector Solitons

Internal modes of vector solitons and their radiation-induced damping are studied analytically and numerically in the framework of coupled nonlinear Schrödinger equations. Bifurcations of internal modes from the integrable systems are analyzed, and the region of their existence in the parameter space of vector solitons is determined. In addition, radiation-induced decay of internal oscillations is investigated. Both exponential and algebraic decay rates are identified.     

1:1内共振条件下矩形薄板的全局分叉和多脉冲混沌动力学

首次利用广义Melnikov方法研究了一个四边简支矩形薄板的全局分叉和多脉冲混沌动力学．矩形薄板受面外的横向激励和面内的参数激励．利用von Krmn模型和Galerkin方法得到一个二自由度非线性非自治系统用来描述矩形薄板的横向振动．在1∶1内共振条件下,利用多尺度方法得到一个四维的平均方程．通过坐标变换把平均方程化为标准形式,利用广义Melnikov方法研究该系统的多脉冲混沌动力学,并且解释了矩形薄板模态间的相互作用机理．在不求同宿轨道解析表达式的前提下,提供了一个计算Melnikov函数的方法．进一步得到了系统的阻尼、激励幅值和调谐参数在满足一定的限制条件下,矩形薄板系统会存在多脉冲混沌运动．数值模拟验证了该矩形薄板的确存在小振幅的多脉冲混沌响应．     

一般符号动力系统的浑沌性态*

本文将符号动力系统理论推广到一般的情况,讨论当X为可分度量空间时,一般符号动力系统(∑(X),σ)及其特例的浑沌性质及应用.     

Dynamics of Embedded Solitons in the Extended Korteweg–de Vries Equations

Embedded solitons are solitary waves residing inside the continuous spectrum of a wave system. They have been discovered in a wide array of physical situations recently. In this article, we present the first comprehensive theory on the dynamics of embedded solitons and nonlocal solitary waves in the framework of the perturbed fifth-order Korteweg–de Vries (KdV) hierarchy equation. Our method is based on the development of a soliton perturbation theory. By obtaining the analytical formula for the tail amplitudes of nonlocal solitary waves, we demonstrate the existence of single-hump embedded solitons for both Hamiltonian and non-Hamiltonian perturbations. These embedded solitons can be isolated (existing at a unique wave speed) or continuous (existing at all wave speeds). Under small wave speed limit, our results show that the tail amplitudes of nonlocal waves are exponentially small, and the product of the amplitude and cosine of the phase is a constant to leading order. This qualitatively reproduces the previous results on the fifth-order KdV equation obtained by exponential asymptotics techniques. We further study the dynamics of embedded solitons and prove that, under Hamiltonian perturbations, a localized wave initially moving faster than the embedded soliton will asymptotically approach this embedded soliton, whereas a localized wave moving slower than the embedded soliton will decay into radiation. Thus, the embedded soliton is semistable. Under non-Hamiltonian perturbations, stable embedded solitons are found for the first time.     

Generation of Solitons in a Ferromagnetic Domain Structure

Computational Mathematics and Mathematical Physics - Generation of solitons in a stripe domain structure of a ferromagnet with an easy-axis anisotropy from an initial localized magnetization pulse...     

Chaotic Dynamics of Monotone Twist Maps

For a monotone twist map, under certain non-degenerate condition, we showed the existence of infinitely many homoclinic and heteroclinic orbits between two periodic neighboring minimal orbits with the same rotation number, which indicates chaotic dynamics. Our results also apply to geodesics of smooth Riemannian metrics on the two-dimension torus.     

Vector Solitons and Their Internal Oscilliations in Birefringent Nonlinear Optical Fibers

In this article, the vector solitons in birefringent nonlinear optical fibers are studied first. Special attention is given to the single-hump vector solitons due to evidences that only they are stable. Questions such as the existence, uniqueness, and total number of these solitons are addressed. It is found that the total number of them is continuously infinite and their polarizations can be arbitrary. Next, the internal oscillations of these vector solitons are investigated by the linearization method. Discrete eigenmodes of the linearized equations are identified. Such modes cause to the vector solitons a kind of permanent internal oscillations, which visually appear to be a combination of translational and width oscillations in the A and B pulses. The numerically observed radiation shelf at the tails of interacting pulses is also explained. Finally, the asymptotic states of the perturbed vector solitons are studied within both the linear and nonlinear theory. It is found that the state of internal oscillations of a vector soliton is always unstable. It invariably emits energy radiation and eventually evolves into a single-hump vector soliton state.     

Generic-Type Results for Chaotic Dynamics in Equations with Periodic Forcing Terms

We prove that a class of equations containing the classical periodically forced pendulum problem displays the main features of chaotic dynamics for a set of forcing terms open and dense in suitable spaces. The approach is based on global variational methods.     

Robust Observer Design for a Class of Nonlinear Systems Using the System Internal Dynamics Structure

In this paper, an observer design strategy is presented for a class of nonlinear systems with structural uncertainty. The modern geometric approach is exploited to simplify the system structure. Then, based on the Lyapunov direct method, a robust observer is proposed using the system internal dynamics structure and the distribution of the uncertainty structure. The bound on the uncertainty, which is employed in the observer design, is allowed to be nonlinear and have a more general form. Simulation shows that the proposed approach is effective.     

Dynamics of Noncommutative Solitons I: Spectral Theory and Dispersive Estimates

We consider the Schrödinger equation with a Hamiltonian given by a second-order difference operator with nonconstant growing coefficients, on the half one-dimensional lattice. This operator appeared first naturally in the construction and dynamics of noncommutative solitons in the context of noncommutative field theory. We prove pointwise in time decay estimates with the decay rate \({t^{-1}\log^{-2}t}\), which is optimal with the chosen weights and appears to be so generally. We use a novel technique involving generating functions of orthogonal polynomials to achieve this estimate.     

Dynamics of a Rolling Disk in the Presence of Dry Friction

In this paper we are interested in the dynamics and numerical treatment of a rolling disk on a flat support. The objective of the paper is to develop a numerical model which is able to simulate the dynamics of a rolling disk taking into account various kinds a friction models (resistance against sliding, pivoting and rolling). A mechanical model of a rolling disk is presented in the framework of Non-smooth Dynamics and Convex Analysis. In an analytical study, approximations are derived for the energy decay of the system during the final stage of the motion for various kinds of frictional dissipation models. Finally, the numerical and analytical results are discussed and compared with experimental results available in literature.     

Dynamics of a Smooth Profile in a Medium with Friction in the Presence of Parametric Excitation

This paper addresses the problem of self-propulsion of a smooth profile in a medium with viscous dissipation and circulation by means of parametric excitation generated by oscillations of the moving internal mass. For the case of zero dissipation, using methods of KAM theory, it is shown that the kinetic energy of the system is a bounded function of time, and in the case of nonzero circulation the trajectories of the profile lie in a bounded region of the space. In the general case, using charts of dynamical regimes and charts of Lyapunov exponents, it is shown that the system can exhibit limit cycles (in particular, multistability), quasi-periodic regimes (attracting tori) and strange attractors. One-parameter bifurcation diagrams are constructed, and Neimark–Sacker bifurcations and period-doubling bifurcations are found. To analyze the efficiency of displacement of the profile depending on the circulation and parameters defining the motion of the internal mass, charts of values of displacement for a fixed number of periods are plotted. A hypothesis is formulated that, when nonzero circulation arises, the trajectories of the profile are compact. Using computer calculations, it is shown that in the case of anisotropic dissipation an unbounded growth of the kinetic energy of the system (Fermi-like acceleration) is possible.     

Optimization of the Internal Dynamics of Underactuated Robots

A robot is underactuated if it possesses less control inputs than degrees of freedom, e.g. due to passive joints. The analysis of the mechanical design of these kinds of underactuated robots often shows that they are non-minimum phase, i.e. they have an internal dynamic which is not asymptotically stable. Therefore, feedback linearization is not possible, and output trajectory tracking becomes a very challenging task. It is shown that through an optimization procedure the mechanical design of an underactuated robot can be altered in such a way that the internal dynamics becomes stable. Thus feedback linearization of the underactuated robot becomes possible. In the optimization procedure, the design parameters are additional masses which are added to defined locations at different un-actuated links of the robot. The optimization criteria is two-stage and firstly requires that all eigenvalues of the linearized zero-dynamics are in the left half-plane and secondly that initial errors in the zero-dynamics decay rapidly. Due to the two-stage criteria computation the optimization problem is discontinuous. Also there might be many local minima. Therefore a particle swarm optimization procedure is used. The efficiency of this optimization approach is demonstrated by simulation of an underactuated robot. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dynamics of Strongly Nonlinear Kinks and Solitons in a Two‐Layer Fluid

Perturbation theory is developed for interaction of strongly nonlinear solitary waves close to the limiting, tabletop solitons (Π‐solitons). The method is based on representing each soliton as a compound of two kinks so that the interaction of N solitons is treated as the interaction of 2N kinks. As an example the Miyata–Choi–Camassa equations for a two‐layer fluid is considered. Equations for kink coordinates are obtained and analyzed. Some nontrivial features of two‐soliton interaction characteristic of the strongly nonlinear case are established.